If \(F(t)\)  is an exponentially decaying function, the \(k\) -th moment of \(\Vert M_\ell \Vert \)  exists for all \(k \in \mathbb {N}\) .
In particular, \(F(t)\)  has the form \(\exp (-t^2)\)  or \(\exp (-t)\) , which corresponds to the sub-Gaussian or sub-exponential property of the distributions ([1]}).
If \(F(t)\)  is a polynomially decaying function \(F(t) = -t^b\)  for some \(b>2\) , the \(k\) -th moment does not exist for some \(k > 2\) , and the form represents the heavier-tail property of \(\Vert M_\ell \Vert \) .
